Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Some numerical results on the characters of exceptional Weyl groups

W. M. Beynona1 and G. Lusztiga2

a1 University of Warwick, Coventry

a2 Mathematics Institute, University of Warwick, and M.I.T

1. Let V be an l-dimensional real vector space and let W be a finite subgroup of GL(V) generated by reflexions such that the space of W-invariant vectors in V is zero. Then W acts naturally on the symmetric algebra S of V preserving the natural grading S0305004100055249_inline1. LetI be the ideal in S generated by the w-invariant elements in S0305004100055249_inline2. The quotient algebra S0305004100055249_inline3 inherits the W-action and also a grading

S0305004100055249_eqnU1

S0305004100055249_inline1 is the image of Sk under S. It is well known that the W-module is isomorphic to the regular representation of W (see (3), ch. v, 5·2); in particular, k = 0 for large k. (More precisely, k = 0 for k > ν, where ν is the number of reflexions in W.)If ρ is an irreducible character of W, we denote by nk(ρ) the multiplicity of ρ in the W- module k. The sequence n(ρ) = (n0(ρ), n1(ρ), n2(ρ), …) is an interesting invariant of the character ρ For example, in the study of unipotent classes in semisimple groups, one encounters the following question: what is the smallest k for which nk(ρ) 4= 0 (with ρ as above) and, then, what is nk(ρ) Also, the polynomial in q

S0305004100055249_eqnU2

can sometimes be interpreted as the dimension of an irreducible representation of a Chevalley group over the field with q elements. For these reasons it seems desirable to describe explicitly the sequence n(ρ) (or, equivalently, the polynomial Pρ(q)) for the various irreducible characters ρ of W. When W is a Weyl group of type Al, this is contained in the work of Steinberg (8); in the case where W is a Weyl group of type Bl or Dl this is done in (7),§2.

(Received March 08 1978)